A337067 Number of nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.

1, 1, 2, 4, 9, 22, 57, 156, 447, 1332, 4103, 12999, 42176, 139638, 470353, 1607861, 5566543, 19484810, 68859862, 245404650, 881081082, 3184214751, 11575346316, 42300703150, 155316289004, 572725968326, 2120154235114, 7876449597257, 29356608044002

Offset: 0

Author

Alois P. Heinz, Oct 12 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1671
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A333069, A333105, A333647, A337318, A337863.

Programs

Maple

b:= proc(x, y) option remember; `if`(x=0, 1, add(add(
      b(x-h, y-v), h=1..min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);

Mathematica

b[x_, y_] := b[x, y] = If[x == 0, 1, Sum[Sum[
    b[x-h, y-v], {h, 1, Min[x-y+v, Max[1, y-v]]}], {v, -1, Min[y, 1]}]];
a[n_] := b[n, 0];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 4^n / n^(3/2), where c = 0.03828240225265266504281697555169550706277641504396262520878537702016362... - Vaclav Kotesovec, Oct 24 2020